1. 445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 4 Compounding the Problem

2. Angle Formulae In this lecture we treat sine, cosine & tangent as mathematical functions which have relationships with each other. These are expressed as various formulae. It is important that you UNDERSTAND this work, but not that you can reproduce it. We would like you to be able to USE the formulae when needed. We want you to become familiar with using cyclic functions in algebraic expressions.

3.  f(x) = sin x  g(x) = A + sin x Vertical shift of A  h(x) = sin(x + A) Horizontal shift of –A  j(x) = sin (Ax) Horizontal squish A times  k(x) = Asin x Vertical stretch A times  m(x) = n(x) sin x Outline shape n(x)

4. Post-Lecture Exercise f(x) = sin (–x) f(x) = cos (–x)

5. Post-Lecture Exercise f(x) = 3sin (2x)f(x) = 2cos ( x / 2 ) f(x) = 2 + sin( x / 3 )

6. Post-Lecture Exercise 3.T(t) = 38.6 + 3sin(πt/8) a)38.6 is the normal temperature b)38.6 + 3sin(πt/8) = 40 3sin(πt/8) = 1.4 sin(πt/8) = 1.4/3 = 0.467 πt/8 = sin -1 (0.467) = 0.486 t = 0.486*8/π = 1.236 after about 1 and a quarter days. 4.Maximum is where sine is minimum i.e. when D = 8 + 2 = 10metres

7. 445.102 Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formulae  Sum and Product Formulae  Summary

8. The Distributive Law 2(a + b) = 2a + 2b (a + b) 2 ≠ a 2 + b 2 = a 2 + 2ab + b 2 (a + b)/2 = a/2 + b/2 log(a + b) ≠ log a + log b = log a. log b sin (a + b) ≠ sin a + sin b = ????????????

9. The Unit Circle Again a sin a b sin b sin (a + b) < sin a + sin b

10. A Graphical Explanation ab(a+b) sin a sin b sin (a+b)

11. 445.102 Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formulae  Sum & Product Formulae  Summary

12. The Formula for 0 ≤ ø ≤ π / 2 a sin a b sin b y x z

13. Lecture 4/5 – Summary Compound Angle Formulae sin (A + B) = sinA.cosB + cosA.sinB sin (A – B) = sinA.cosB – cosA.sinB cos (A + B) = cosA.cosB – sinA.sinB cos (A – B) = cosA.cosB + sinA.sinB tan (A + B) = (tanA + tanB) 1 – tanA.tanB tan (A – B) = (tanA – tanB) 1 + tanA.tanB

14. Shelter from the Storm 7m 4m ø 4 cosø + 7sinø

15. Shelter from the Storm 7m 4m ø 7 µ 4 √65 4 cosø + 7sinø

16. Shelter from the Storm 7m 4m ø 4 cosø + 7sinø 7 µ 4 √65 sinµ = 4/√65 cosµ = 7/√65 4 = √65 sinµ 7 = √65 cosµ

17. Shelter from the Storm 7m 4m ø √65sinµ cosø + √65cosµsinø 7 µ 4 √65 sinµ = 4/√65 cosµ = 7/√65 4 = √65 sinµ 7 = √65 cosµ

18. 445.102 Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formula  Sum & Product Formulae  Summary

19. Double Angle Formulae sin (A + B) = sinA.cosB + cosA.sinB sin 2A = sinA.cosA + cosA.sinA = 2sinA cosA cos (A + B) = cosA.cosB – sinA.sinB cos 2A = cosA.cosA – sinA.sinA = cos 2 A – sin 2 A

20. Double Angle Formulae tan (A + B) = (tanA + tanB) 1 – tanA.tanB tan 2A = (tanA + tanA) 1 – tanA.tanA tan 2A = 2tanA 1 – tan 2 A

21. 445.102 Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formula  Sum & Product Formulae  Summary

22. The Octopus Large wheel, radius 6m, 8 second period. A = 6sin(2πx/8)

23. The Octopus Add a small wheel, radius 1.5m, 2s period. B = 1.5sin(2πx/2)

24. The Octopus Combine the two...... A + B = 6sin(2πx/8) + 1.5sin(2πx/2)

25. The Surf Decent surf has a height of 1.5m, 15s period. A = 1.5sin(2πx/15)

26. The Surf Add similar wave, say: 1m, 13s period. A + B = 1.5sin(2πx/15) + 1sin(2πx/13)

27. Adding Sine Functions sin(A+B) = sinAcosB + sinBcosA sin(A–B) = sinAcosB – sinBcosA Adding......... sin(A+B) + sin(A–B) = 2sinAcosB Rearranging......... sinAcosB = 1 / 2 [sin(A+B) + sin(A–B)]

28. Adding Sine Functions sinAcosB = 1 / 2 [sin(A+B) + sin(A–B)] Or, making A = (P+Q) / 2 and B = (P–Q) / 2 That is: A+B = 2P / 2 and A–B = 2Q / 2 1 / 2 [sin P + sin Q] = sin (P+Q) / 2 cos (P–Q) / 2 sin P + sin Q = 2 sin (P+Q) / 2 cos (P–Q) / 2

29. 445.102 Lecture 4/4  Administration  Last Lecture  Distributive Functions  Explanations of sin(A + B)  Developing a Formula  Further Formulae  Summary

30. Lecture 4/4 – Summary Compounding the Problem Please KNOW THAT these formulae exist Please BE ABLE to follow the logic of their derivation and use Please PRACTISE the simple applications of the formulae as in the Post-Lecture exercises